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mo_functions.f90
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mo_functions.f90
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!> \file mo_functions.f90
!> \brief Special functions such as gamma function.
!> \details This module provides special functions such as the gamma function.
!> \authors Matthias Cuntz
!> \date May 2014
MODULE mo_functions
! Provide special functions.
! Written Matthias Cuntz, May 2014
! Modified Matthias Cuntz, Dec 2017 - morris
! Stephan Thober, Aug 2020 - added beta function
! License
! -------
! This file is part of the JAMS Fortran package, distributed under the MIT License.
!
! Copyright (c) 2014-2017 Matthias Cuntz - mc (at) macu (dot) de
!
! Permission is hereby granted, free of charge, to any person obtaining a copy
! of this software and associated documentation files (the "Software"), to deal
! in the Software without restriction, including without limitation the rights
! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
! copies of the Software, and to permit persons to whom the Software is
! furnished to do so, subject to the following conditions:
!
! The above copyright notice and this permission notice shall be included in all
! copies or substantial portions of the Software.
!
! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
! SOFTWARE.
USE mo_kind, ONLY: i4, i8, sp, dp
IMPLICIT NONE
PUBLIC :: factln ! ln(n!)
PUBLIC :: factorial ! n!
PUBLIC :: gammln ! ln(gamma)
PUBLIC :: gamm ! gamm
PUBLIC :: morris ! elementary effects test function after Morris (1991)
PUBLIC :: beta ! beta function
! ------------------------------------------------------------------
!
! NAME
! factln
!
! PURPOSE
! Logarithm of factorial.
!
!> \brief \f$ \ln(n!) \f$.
!
!> \details Elemental function of the logarithm of the factorial:
!> \f[ \ln(n!) \f]
!> with
!> \f[ n! = 1 \dot 2 \dot ... \dot n \f]
!>
!> Uses n! = \f$ \Gamma(n+1) \f$.
!
! INTENT(IN)
!> \param[in] "integer(i4/i8) :: n" input
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(dp) :: factln — \f$ \ln(n!) \f$
!
! RESTRICTIONS
!> \note return is always double precision.
!
! EXAMPLE
! fac = exp(factln(5))
! fac -> 120
! -> see also example in test directory
!
! HISTORY
!> \author Matthias Cuntz
!> \date May 2014
INTERFACE factln
MODULE PROCEDURE factln_i4, factln_i8
END INTERFACE factln
! ------------------------------------------------------------------
!
! NAME
! factorial
!
! PURPOSE
! Factorial.
!
!> \brief \f$ n! \f$.
!
!> \details Elemental function of the factorial:
!> \f[ n! = 1 \dot 2 \dot ... \dot n \f]
!>
!> Uses n! = \f$ \Gamma(n+1) \f$.
!
! INTENT(IN)
!> \param[in] "integer(i4/i8) :: n" input
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(dp) :: factln — \f$ \ln(n!) \f$
!
! RESTRICTIONS
!> \note return is always double precision.
!
! EXAMPLE
! fac = factorial(5)
! fac -> 120
! -> see also example in test directory
!
! HISTORY
!> \author Matthias Cuntz
!> \date Feb 2013
! ------------------------------------------------------------------
INTERFACE factorial
MODULE PROCEDURE factorial_i4, factorial_i8
END INTERFACE factorial
! ------------------------------------------------------------------
!
! NAME
! gamm
!
! PURPOSE
! The gamma function.
!
!> \brief \f$ \Gamma(z) \f$.
!
!> \details Elemental function of the gamma function:
!> \f[ \Gamma = \Int_0^\Inf t^{z-1} e^{-t} dt \f]
!>
!> Uses Lanczos-type approximation to ln(gamma) for z > 0.
!> The function calculates gamma from abs(z).
!> Accuracy: About 14 significant digits except for small regions in the vicinity of 1 and 2.
!
!
! INTENT(IN)
!> \param[in] "real(sp/dp) :: z" input
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(sp/dp) :: gamm — \f$ \Gamma(z) \f$
!
! RESTRICTIONS
!> \note z must be >0. gammln is calculated from abs(z).
!
! EXAMPLE
! vec = (/ 1., 2, 3., -9., 5., 6. /)
! gl = gamm(vec)
! -> see also example in test directory
!
! LITERATURE
! Lanczos, C. ''A precision approximation of the gamma function'', J. SIAM Numer. Anal., B, 1, 86-96, 1964.
!
! HISTORY
!> \author Alan Miller, 1 Creswick Street, Brighton, Vic. 3187, Australia, e-mail: amiller @ bigpond.net.au
!> \date Oct 1996
! Modified, Matthias Cuntz, May 2014
INTERFACE gamm
MODULE PROCEDURE gamm_sp, gamm_dp
END INTERFACE gamm
! ------------------------------------------------------------------
!
! NAME
! gammln
!
! PURPOSE
! Logarithm of the gamma function.
!
!> \brief \f$ \ln(\Gamma(z)) \f$.
!
!> \details Elemental function of the logarithm of the gamma function:
!> \f[ \Gamma = \Int_0^\Inf t^{z-1} e^{-t} dt \f]
!>
!> Uses Lanczos-type approximation to ln(gamma) for z > 0.
!> The function calculates ln(gamma) from abs(z).
!> Accuracy: About 14 significant digits except for small regions in the vicinity of 1 and 2.
!
!
! INTENT(IN)
!> \param[in] "real(sp/dp) :: z" input
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(sp/dp) :: gammln — \f$ \ln\Gamma(z) \f$
!
! RESTRICTIONS
!> \note z must be >0. gammln is calculated from abs(z).
!
! EXAMPLE
! vec = (/ 1., 2, 3., -9., 5., 6. /)
! gl = gammln(vec)
! -> see also example in test directory
!
! LITERATURE
! Lanczos, C. ''A precision approximation of the gamma function'', J. SIAM Numer. Anal., B, 1, 86-96, 1964.
!
! HISTORY
!> \author Alan Miller, 1 Creswick Street, Brighton, Vic. 3187, Australia, e-mail: amiller @ bigpond.net.au
!> \date Oct 1996
! Modified, Matthias Cuntz, May 2014
INTERFACE gammln
MODULE PROCEDURE gammln_sp, gammln_dp
END INTERFACE gammln
! ------------------------------------------------------------------
!
! NAME
! morris
!
! PURPOSE
!> \details 20-dimension test function for elementary effects after Morris (1991).
!
! INTENT(IN)
!> \param[in] "real(sp/dp), dimension(20[,npoints]) :: x" number of points
!> \param[in] "real(sp/dp) :: beta0" parameters
!> \param[in] "real(sp/dp), dimension(20) :: beta1" parameters
!> \param[in] "real(sp/dp), dimension(20,20) :: beta2" parameters
!> \param[in] "real(sp/dp), dimension(20,20,20) :: beta3" parameters
!> \param[in] "real(sp/dp), dimension(20,20,20,20) :: beta4" parameters
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(sp/dp)[, dimension(npoints)] :: morris — result of Morris function
!
! RESTRICTIONS
! None
!
! EXAMPLE
! -> see example in test directory
!
! LITERATURE
! Morris (1991), Factorial sampling plans for preliminary computational experiments,
! Technometrics 33, 161-174.
!
! HISTORY
!> \author Matthias Cuntz, Juliane Mai, Mar 2015 in Python
!> \date Mar 2015
! Modified, Matthias Cuntz, Dec 2017 - ported to Fortran
INTERFACE morris
MODULE PROCEDURE morris_0d_dp, morris_1d_dp, &
morris_0d_sp, morris_1d_sp
END INTERFACE morris
! ------------------------------------------------------------------
!
! NAME
! beta
!
! PURPOSE
!> \details calculates beta function
!
! INTENT(IN)
!> \param[in] "real(sp/dp) :: alpha" parameters
!> \param[in] "real(sp/dp) :: beta" parameters
!
! INTENT(INOUT)
! None
!
! INTENT(OUT)
! None
!
! INTENT(IN), OPTIONAL
! None
!
! INTENT(INOUT), OPTIONAL
! None
!
! INTENT(OUT), OPTIONAL
! None
!
! RETURNS
!> \return real(sp/dp)[, dimension(npoints)] :: beta — result of beta function
!
! RESTRICTIONS
! None
!
! EXAMPLE
! -> see example in test directory
!
! LITERATURE
! https://en.wikipedia.org/wiki/Beta_function
!
! HISTORY
!> \author Stephan Thober
!> \date Aug 2020
! Modified
INTERFACE beta
MODULE PROCEDURE beta_sp, beta_dp
END INTERFACE beta
! ------------------------------------------------------------------
PRIVATE
! ------------------------------------------------------------------
CONTAINS
! ------------------------------------------------------------------
ELEMENTAL PURE FUNCTION factln_i4(n)
IMPLICIT NONE
INTEGER(i4), INTENT(IN) :: n
REAL(dp) :: factln_i4
factln_i4 = gammln(real(n,dp)+1.0_dp)
end function factln_i4
ELEMENTAL PURE FUNCTION factln_i8(n)
IMPLICIT NONE
INTEGER(i8), INTENT(IN) :: n
REAL(dp) :: factln_i8
factln_i8 = gammln(real(n,dp)+1.0_dp)
end function factln_i8
! ------------------------------------------------------------------
elemental pure function factorial_i4(n)
! Returns the factorial n!
IMPLICIT NONE
INTEGER(i4), INTENT(IN) :: n
INTEGER(i4) :: factorial_i4
factorial_i4 = nint(exp(factln(n)),i4)
end function factorial_i4
elemental pure function factorial_i8(n)
! Returns the factorial n!
IMPLICIT NONE
INTEGER(i8), INTENT(IN) :: n
INTEGER(i8) :: factorial_i8
factorial_i8 = nint(exp(factln(n)),i8)
end function factorial_i8
! ------------------------------------------------------------------
elemental pure function gamm_dp(z)
IMPLICIT NONE
REAL(dp), INTENT(IN) :: z
REAL(dp) :: gamm_dp
gamm_dp = exp(gammln(z))
end function gamm_dp
elemental pure function gamm_sp(z)
IMPLICIT NONE
REAL(sp), INTENT(IN) :: z
REAL(sp) :: gamm_sp
gamm_sp = exp(gammln(z))
end function gamm_sp
! ------------------------------------------------------------------
ELEMENTAL PURE FUNCTION gammln_dp(z)
! Uses Lanczos-type approximation to ln(gamma) for z > 0.
! Reference:
! Lanczos, C. ''A precision approximation of the gamma
! function'', J. SIAM Numer. Anal., B, 1, 86-96, 1964.
! Accuracy: About 14 significant digits except for small regions
! in the vicinity of 1 and 2.
! Programmer: Alan Miller
! 1 Creswick Street, Brighton, Vic. 3187, Australia
! e-mail: amiller @ bigpond.net.au
! Latest revision - 14 October 1996
IMPLICIT NONE
REAL(dp), INTENT(IN) :: z
REAL(dp) :: gammln_dp
! Local variables
REAL(dp), parameter :: a(9) = (/ 0.9999999999995183_dp, 676.5203681218835_dp, &
-1259.139216722289_dp, 771.3234287757674_dp, &
-176.6150291498386_dp, 12.50734324009056_dp, &
-0.1385710331296526_dp, 0.9934937113930748E-05_dp, &
0.1659470187408462E-06_dp /)
REAL(dp), parameter :: zero = 0.0_dp, &
one = 1.0_dp, &
lnsqrt2pi = 0.9189385332046727_dp, &
half = 0.5_dp, &
sixpt5 = 6.5_dp, &
seven = 7.0_dp
REAL(dp) :: tmp
INTEGER :: j
! if (z <= 0.0_dp) stop 'Error gammln_dp: z <= 0'
gammln_dp = zero
tmp = abs(z) + seven
DO j = 9, 2, -1
gammln_dp = gammln_dp + a(j)/tmp
tmp = tmp - one
END DO
gammln_dp = gammln_dp + a(1)
gammln_dp = LOG(gammln_dp) + lnsqrt2pi - (abs(z) + sixpt5) + (abs(z) - half)*LOG(abs(z) + sixpt5)
END FUNCTION gammln_dp
ELEMENTAL PURE FUNCTION gammln_sp(z)
! Uses Lanczos-type approximation to ln(gamma) for z > 0.
! Reference:
! Lanczos, C. ''A precision approximation of the gamma
! function'', J. SIAM Numer. Anal., B, 1, 86-96, 1964.
! Accuracy: About 14 significant digits except for small regions
! in the vicinity of 1 and 2.
! Programmer: Alan Miller
! 1 Creswick Street, Brighton, Vic. 3187, Australia
! e-mail: amiller @ bigpond.net.au
! Latest revision - 14 October 1996
IMPLICIT NONE
REAL(sp), INTENT(IN) :: z
REAL(sp) :: gammln_sp
! Local variables
REAL(sp), parameter :: a(9) = (/ 0.9999999999995183_sp, 676.5203681218835_sp, &
-1259.139216722289_sp, 771.3234287757674_sp, &
-176.6150291498386_sp, 12.50734324009056_sp, &
-0.1385710331296526_sp, 0.9934937113930748E-05_sp, &
0.1659470187408462E-06_sp /)
REAL(sp), parameter :: zero = 0.0_sp, &
one = 1.0_sp, &
lnsqrt2pi = 0.9189385332046727_sp, &
half = 0.5_sp, &
sixpt5 = 6.5_sp, &
seven = 7.0_sp
REAL(sp) :: tmp
INTEGER :: j
! if (z <= 0.0_sp) stop 'Error gammln_sp: z <= 0'
gammln_sp = zero
tmp = abs(z) + seven
DO j = 9, 2, -1
gammln_sp = gammln_sp + a(j)/tmp
tmp = tmp - one
END DO
gammln_sp = gammln_sp + a(1)
gammln_sp = LOG(gammln_sp) + lnsqrt2pi - (abs(z) + sixpt5) + (abs(z) - half)*LOG(abs(z) + sixpt5)
END FUNCTION gammln_sp
! ------------------------------------------------------------------
FUNCTION morris_0d_dp(x, beta0, beta1, beta2, beta3, beta4)
use mo_kind, only: dp, i4
implicit none
real(dp), dimension(20), intent(in) :: x
real(dp), intent(in) :: beta0
real(dp), dimension(20), intent(in) :: beta1
real(dp), dimension(20,20), intent(in) :: beta2
real(dp), dimension(20,20,20), intent(in) :: beta3
real(dp), dimension(20,20,20,20), intent(in) :: beta4
real(dp) :: morris_0d_dp
! local
real(dp), dimension(20) :: om
integer(i4), dimension(3) :: ii
integer(i4) :: i, j, l, s, nn
om = 2._dp*(x - 0.5_dp)
ii = (/2, 4, 6/) + 1 ! +1 Python -> Fortran
om(ii) = 2._dp*(1.1_dp*x(ii)/(x(ii)+0.1_dp) - 0.5_dp)
nn = size(x)
morris_0d_dp = beta0
do i=1, nn
morris_0d_dp = morris_0d_dp + beta1(i)*om(i)
do j=i+1, nn
morris_0d_dp = morris_0d_dp + beta2(i,j)*om(i)*om(j)
do l=j+1, nn
morris_0d_dp = morris_0d_dp + beta3(i,j,l)*om(i)*om(j)*om(l)
do s=l+1, nn
morris_0d_dp = morris_0d_dp + beta4(i,j,l,s)*om(i)*om(j)*om(l)*om(s)
end do
end do
end do
end do
END FUNCTION morris_0d_dp
FUNCTION morris_1d_dp(x, beta0, beta1, beta2, beta3, beta4)
use mo_kind, only: dp, i4
implicit none
real(dp), dimension(:,:), intent(in) :: x
real(dp), intent(in) :: beta0
real(dp), dimension(20), intent(in) :: beta1
real(dp), dimension(20,20), intent(in) :: beta2
real(dp), dimension(20,20,20), intent(in) :: beta3
real(dp), dimension(20,20,20,20), intent(in) :: beta4
real(dp), dimension(size(x,2)) :: morris_1d_dp
! local
real(dp), dimension(20,size(x,2)) :: om
integer(i4), dimension(3) :: ii
integer(i4) :: i, j, l, s, nn
om = 2._dp*(x - 0.5_dp)
ii = (/2, 4, 6/)
om(ii,:) = 2._dp*(1.1_dp*x(ii,:)/(x(ii,:)+0.1_dp) - 0.5_dp)
nn = size(x)
morris_1d_dp(:) = beta0
do i=1, nn
morris_1d_dp(:) = morris_1d_dp(:) + beta1(i)*om(i,:)
do j=i+1, nn
morris_1d_dp(:) = morris_1d_dp(:) + beta2(i,j)*om(i,:)*om(j,:)
do l=j+1, nn
morris_1d_dp(:) = morris_1d_dp(:) + beta3(i,j,l)*om(i,:)*om(j,:)*om(l,:)
do s=l+1, nn
morris_1d_dp(:) = morris_1d_dp(:) + beta4(i,j,l,s)*om(i,:)*om(j,:)*om(l,:)*om(s,:)
end do
end do
end do
end do
END FUNCTION morris_1d_dp
FUNCTION morris_0d_sp(x, beta0, beta1, beta2, beta3, beta4)
use mo_kind, only: sp, i4
implicit none
real(sp), dimension(20), intent(in) :: x
real(sp), intent(in) :: beta0
real(sp), dimension(20), intent(in) :: beta1
real(sp), dimension(20,20), intent(in) :: beta2
real(sp), dimension(20,20,20), intent(in) :: beta3
real(sp), dimension(20,20,20,20), intent(in) :: beta4
real(sp) :: morris_0d_sp
! local
real(sp), dimension(20) :: om
integer(i4), dimension(3) :: ii
integer(i4) :: i, j, l, s, nn
om = 2._sp*(x - 0.5_sp)
ii = (/2, 4, 6/)
om(ii) = 2._sp*(1.1_sp*x(ii)/(x(ii)+0.1_sp) - 0.5_sp)
nn = size(x)
morris_0d_sp = beta0
do i=1, nn
morris_0d_sp = morris_0d_sp + beta1(i)*om(i)
do j=i+1, nn
morris_0d_sp = morris_0d_sp + beta2(i,j)*om(i)*om(j)
do l=j+1, nn
morris_0d_sp = morris_0d_sp + beta3(i,j,l)*om(i)*om(j)*om(l)
do s=l+1, nn
morris_0d_sp = morris_0d_sp + beta4(i,j,l,s)*om(i)*om(j)*om(l)*om(s)
end do
end do
end do
end do
END FUNCTION morris_0d_sp
FUNCTION morris_1d_sp(x, beta0, beta1, beta2, beta3, beta4)
use mo_kind, only: sp, i4
implicit none
real(sp), dimension(:,:), intent(in) :: x
real(sp), intent(in) :: beta0
real(sp), dimension(20), intent(in) :: beta1
real(sp), dimension(20,20), intent(in) :: beta2
real(sp), dimension(20,20,20), intent(in) :: beta3
real(sp), dimension(20,20,20,20), intent(in) :: beta4
real(sp), dimension(size(x,2)) :: morris_1d_sp
! local
real(sp), dimension(20,size(x,2)) :: om
integer(i4), dimension(3) :: ii
integer(i4) :: i, j, l, s, nn
om = 2._sp*(x - 0.5_sp)
ii = (/2, 4, 6/)
om(ii,:) = 2._sp*(1.1_sp*x(ii,:)/(x(ii,:)+0.1_sp) - 0.5_sp)
nn = size(x)
morris_1d_sp(:) = beta0
do i=1, nn
morris_1d_sp(:) = morris_1d_sp(:) + beta1(i)*om(i,:)
do j=i+1, nn
morris_1d_sp(:) = morris_1d_sp(:) + beta2(i,j)*om(i,:)*om(j,:)
do l=j+1, nn
morris_1d_sp(:) = morris_1d_sp(:) + beta3(i,j,l)*om(i,:)*om(j,:)*om(l,:)
do s=l+1, nn
morris_1d_sp(:) = morris_1d_sp(:) + beta4(i,j,l,s)*om(i,:)*om(j,:)*om(l,:)*om(s,:)
end do
end do
end do
end do
END FUNCTION morris_1d_sp
! ------------------------------------------------------------------
elemental pure function beta_sp(alpha, beta)
use mo_kind, only: sp
real(sp), intent(in) :: alpha
real(sp), intent(in) :: beta
real(sp) :: beta_sp
beta_sp = (gamm(alpha) * gamm(beta)) / gamm(alpha + beta)
end function beta_sp
elemental pure function beta_dp(alpha, beta)
use mo_kind, only: dp
real(dp), intent(in) :: alpha
real(dp), intent(in) :: beta
real(dp) :: beta_dp
beta_dp = (gamm(alpha) * gamm(beta)) / gamm(alpha + beta)
end function beta_dp
END MODULE mo_functions